Free energy is a fundamental molecular property that plays an important role in characterizing chemical and biological systems. An understanding of the free energy behavior of many chemical and biochemical processes, such as protein-ligand binding, can be of importance in endeavors such as rational drug design (which involves the design of small molecules that bind to a biomolecular target).
Computer modeling and simulations are often used in free energy studies. In most instances, evaluation of accurate absolute free energies from simulations is extremely difficult, if at all possible. Hence, the free energy difference between two well-delineated thermodynamic states, or relative free energy, are often used as a study system to provide insight to particular systems, such as a relative binding affinity of a ligand predicated on the measured affinity of a different but similar ligand (e.g., a congeneric ligand).
In the relative free energy calculations, the two thermodynamic states can be referred to as a reference molecule and a target molecule, which can represent respectively an initial state of a molecular system, such as a first molecule, and an ending state of the molecule after one or more transformations have taken place (such as a conformational change, topological change, or a replacement of one atom or chemical group with another (i.e., a mutation)). The term “molecule” can refer to both neutral and charged species. Such transformations may not always represent realistic physical transformations, but may involve nonphysical or “alchemical” transformations. Different frameworks have been developed for calculating free energy differences, such as free energy perturbations (FEP), thermodynamic integrations (TI), and umbrella sampling.
Within the FEP framework, the free energy difference ΔFa→b between the two molecules a and b can be expressed by:
                              Δ          ⁢                                          ⁢                      F                          a              →              b                                      =                              -                          1              β                                ⁢          ln          ⁢                                    〈                              exp                ⁢                                  {                                      -                                          β                      ⁡                                              [                                                                                                            ℋ                              b                                                        ⁡                                                          (                                                              x                                ,                                                                  p                                  x                                                                                            )                                                                                -                                                                                    ℋ                              a                                                        ⁡                                                          (                                                              x                                ,                                                                  p                                  x                                                                                            )                                                                                                      ]                                                                              }                                            〉                        a                                              (        1        )            where β−1=kBT, whereis kB is the Boltzmann constant, T is the temperature. a(x,px) and b(x,px) are the Hamiltonians characteristic of states a and b respectively. ( . . . ) a denotes an ensemble average over configurations representative of the initial, reference molecule, a.
In practical applications of FEP, the transformation between the two thermodynamic states is usually achieved by a series of transformations between non-physical, transition states along a well-delineated pathway that connects a to b. This pathway is often characterized by a general extent parameter, often referred to as a coupling parameter, λ, which varies from 0 to 1 from the reference molecule to the target molecule, and relates the Hamiltonians of the two states by:(λ)=(1−λ)Ha+λHb  (2)where H(λ) is the λ-coupled or hybrid Hamiltonian of the system between the two states (including the two states, when λ takes the end values of 0 and 1). Hence, the free energy difference ΔFa→b between a and b will be:
                                                                        Δ                ⁢                                                                  ⁢                                  F                                      a                    →                    b                                                              =                            ⁢                                                -                                      1                    β                                                  ⁢                ln                ⁢                                                      〈                                          exp                      ⁢                                              {                                                  -                                                      β                            ⁡                                                          [                                                                                                ℋ                                  ⁡                                                                      (                                                                          λ                                      =                                      1                                                                        )                                                                                                  -                                                                  ℋ                                  ⁡                                                                      (                                                                          λ                                      =                                      0                                                                        )                                                                                                                              ]                                                                                                      }                                                              〉                                                        λ                    =                    0                                                                                                                          =                            ⁢                                                -                                      1                    β                                                  ⁢                                                      ∑                                          i                      =                      0                                                              N                      -                      1                                                        ⁢                                                                          ⁢                                      ln                    ⁢                                                                  〈                                                  exp                          ⁢                                                      {                                                          -                                                              β                                ⁡                                                                  [                                                                                                            ℋ                                      ⁡                                                                              (                                                                                  x                                          ,                                                                                                                                    p                                              x                                                                                        ;                                                                                          λ                                                                                              i                                                +                                                1                                                                                                                                                                                                                    )                                                                                                              -                                                                          ℋ                                      ⁡                                                                              (                                                                                  x                                          ,                                                                                                                                    p                                              x                                                                                        ;                                                                                          λ                                              i                                                                                                                                                                      )                                                                                                                                              ]                                                                                                                      }                                                                          〉                                            i                                                                                                                              (        3        )            where N stands for the number of “windows” between neighboring states between the reference (initial) state and the target (final) state, and λi is the values of the coupling parameter in the initial, intermediate, and final state.
The free energy difference between the reference system state a and the target system state b can also be calculated using thermodynamic integration method, where the free energy difference is calculated using the following formula:
                              Δ          ⁢                                          ⁢                      F                          a              →              b                                      =                              ∫                          λ              =              0                                      λ              =              1                                ⁢                      d            ⁢                                                  ⁢            λ            ⁢                                          〈                                                      ∂                                          ℋ                      ⁡                                              (                        λ                        )                                                                                                  ∂                    λ                                                  〉                            λ                                                          (        4        )            where λ is the coupling parameter which varies from 0 to 1 from the reference state to the target molecule, H(λ) is the λ-coupled or hybrid Hamiltonian of the system between the two states (including the two states, when λ takes the end values of 0 and 1), and
      ∂          ℋ      ⁡              (        λ        )                  ∂    λ  is the first derivative of the coupled Hamiltonian with respect to the coupling parameter λ. In practical applications of TI, the transformation between the reference system state and the target system state is achieved by a series transformations along a well-delineated pathway that connects a to b, and the ensemble average of
      ∂          ℋ      ⁡              (        λ        )                  ∂    λ  is calculated for all the states sampled, including the reference system state, the intermediate non-physical states, and the target system state. The free energy difference between the reference system state and the target system state is then approximated by numerical integration of the above integral based on the value of the
            〈                        ∂                      ℋ            ⁡                          (              λ              )                                                ∂          λ                    〉              λ      i        ,where λi is the values of the coupling parameter in the initial, intermediate, and final states.